Consider a static spherically-symmetric metric in the following form

where is the metric on the -dimensional unit sphere, describing a non-extreme hole with an outer horizon located at . However, the analysis can be made for a more general metric, in which , the limiting geometry is the simplest in the case we consider in Eq. (243). The function in Eq. (243) can be expanded as follows It is convenient to consider the geodesic distance as a radial coordinate. Retaining the first two terms in Eq. (244), we find, for , that In order to avoid the appearance of a conical singularity at , the Euclidean time in Eq. (243) must be compactified with period , which goes to infinity in the extreme limit . However, rescaling yields a new variable having period . Then, taking into account Eq. (245), one finds for the metric (243) where we have introduced the variable . To obtain the extremal limit one just takes . The limiting geometry is that of the direct product of a 2-dimensional space and a -sphere and is characterized by a pair of dimensional parameters and . The parameter sets the radius of the -dimensional sphere, while the parameter is the curvature radius for the 2-space. Clearly, this two-dimensional space is the negative constant curvature space . This is the universality we mentioned above: although the non-extreme geometry is in general described by an infinite number of parameters associated with the determining function , the geometry in the extreme limit depends only on two parameters and . Note that the coordinate is inadequate for describing the extremal limit (247) since the coordinate transformation (245) is singular when . The limiting metric (247) is characterized by a finite temperature, determined by the periodicity in angular coordinate .The limiting geometry (247) is that of a direct product of 2d hyperbolic space with radius and a 2D sphere with radius . It is worth noting that the limiting geometry (247) precisely merges near the horizon with the geometry of the original metric (243) in the sense that all the curvature tensors for both metrics coincide. This is in contrast with, say, the situation in which the Rindler metric is considered to approximate the geometry of a non-extreme black hole: the curvatures of both spaces do not merge in general.

For a special type of extremal black holes , the limiting geometry is characterized by just one dimensionful parameter. This is the case for the Reissner–Nordström black hole in four dimensions. The limiting extreme geometry in this case is the well-known Bertotti–Robinson space characterized by just one parameter . This space has remarkable properties in the context of supergravity theory that are not the subject of the present review.

Consider now a scalar field propagating on the background of the limiting geometry (247) and described by the operator

where is the Ricci scalar. For the metric (247) characterized by two dimensionful parameters and , one has that . For a -dimensional conformally-coupled scalar field we have . In this caseThe calculation of the respective entanglement entropy goes along the same lines as before. First, one allows the coordinate , which plays the role of the Euclidean time, to have period . For the metric (247) then describes the space , where is the hyperbolic space coinciding with everywhere except the point , where it has a conical singularity with an angular deficit . The heat kernel of the Laplace operator on is given by the product

Let us denote the trace of the heat kernel of the Laplace operator on a -dimensional sphere of unit radius. The entanglement entropy in the extremal limit then takes the form [172, 206]

The function has the following small- expansion The trace of the heat kernel on a sphere is known in some implicit form. However, for our purposes a representation in a form of an expansion is more useful, where is the area of a unit radius sphere . The first few coefficients in this expansion can be calculated using the results collected in [219],We shall consider some particular cases.

The entropy takes the form:

Living Rev. Relativity 14, (2011), 8
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